Power Series Expansion for the Time Evolution Operator with a Harmonic-Oscillator Reference System

A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schrödinger equations is presented. The main idea is representing the propagator in the form $P(x, t|x_0)=P_0(x, t\left|x_0\right)\exp \left[W\right(x, t\left|x_0\right)]$ and expanding the exponent $W$ in a power series in a given function of $t$, where $P_0$ is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.